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Macaulay2Doc :: solve

solve -- solve a linear equation

Synopsis

Description

(Disambiguation: for division of matrices, which can also be thought of as solving a system of linear equations, see instead Matrix // Matrix. For lifting a map between modules to a map between their free resolutions, see extend.)

There are several restrictions. The first is that there are only a limited number of rings for which this function is implemented. Second, over RR or CC, the matrix A must be a square non-singular matrix. Third, if A and b are mutable matrices over RR or CC, they must be dense matrices.
i1 : kk = ZZ/101;
i2 : A = matrix"1,2,3,4;1,3,6,10;19,7,11,13" ** kk

o2 = | 1  2 3  4  |
     | 1  3 6  10 |
     | 19 7 11 13 |

              3        4
o2 : Matrix kk  <--- kk
i3 : b = matrix"1;1;1" ** kk

o3 = | 1 |
     | 1 |
     | 1 |

              3        1
o3 : Matrix kk  <--- kk
i4 : x = solve(A,b)

o4 = | 2  |
     | -1 |
     | 34 |
     | 0  |

              4        1
o4 : Matrix kk  <--- kk
i5 : A*x-b

o5 = 0

              3        1
o5 : Matrix kk  <--- kk
Over RR or CC, the matrix A must be a non-singular square matrix.
i6 : printingPrecision = 2;
i7 : A = matrix "1,2,3;1,3,6;19,7,11" ** RR

o7 = | 1  2 3  |
     | 1  3 6  |
     | 19 7 11 |

                3          3
o7 : Matrix RR    <--- RR
              53         53
i8 : b = matrix "1;1;1" ** RR

o8 = | 1 |
     | 1 |
     | 1 |

                3          1
o8 : Matrix RR    <--- RR
              53         53
i9 : x = solve(A,b)

o9 = | -.15 |
     | 1.1  |
     | -.38 |

                3          1
o9 : Matrix RR    <--- RR
              53         53
i10 : A*x-b

o10 = | 2.2e-16  |
      | -2.2e-16 |
      | 0        |

                 3          1
o10 : Matrix RR    <--- RR
               53         53
i11 : norm oo

o11 = 2.22044604925031e-16

o11 : RR (of precision 53)
For large dense matrices over RR or CC, this function calls the lapack routines.
i12 : n = 10;
i13 : A = random(CC^n,CC^n)

o13 = | .37+.1i   .86+.93i .74+.08i  .33+.29i  .64+.85i .49+.3i  .57+.25i 
      | .21+.018i .79+.62i .35+.23i  .51+.57i  .93+.92i .77+.77i .87+.8i  
      | .21+.96i  .63+.99i .48+.57i  .7+.3i    .57+.96i .57+.77i .78+.34i 
      | .98+.45i  .04+.39i .08+.84i  .096+.44i .71+.24i .82+.07i .76+.47i 
      | .37+.28i  .81+.16i .2+.56i   .65+.37i  .87+i    .79+.3i  .02+.98i 
      | .84+.05i  .97+.26i .91+.47i  .41+.52i  .44+.85i .34+.94i .24+.022i
      | .97+.14i  .47+.62i .15+.72i  .12+.43i  .24+.52i .27+.38i .44+.65i 
      | .43+.54i  .49+.57i .97+.01i  .98+.46i  .48+.69i .27+.52i .96+.04i 
      | .16+.97i  .72+.46i .36+.74i  .89+.8i   .94+.63i .33+.47i .08+.67i 
      | .026+.24i .56+.05i .027+.36i .46+.77i  .37+.27i .9+.64i  .39+.42i 
      -----------------------------------------------------------------------
      .51+.4i   .95+.62i .37+.51i  |
      .83+.74i  .06+.97i .73+.29i  |
      .88+.45i  .67+.72i .18+.043i |
      .35+.69i  .63+.65i .93+.99i  |
      .85+.88i  .68+.47i .76+.13i  |
      .82+.92i  .51+.96i .92+.93i  |
      .65+.81i  .58+.27i .51+.21i  |
      .021+.24i .41+.95i .47+.58i  |
      .1+.55i   .7+.56i  .43+.1i   |
      .09+.093i .84+.8i  .68+.59i  |

                 10          10
o13 : Matrix CC     <--- CC
               53          53
i14 : b = random(CC^n,CC^2)

o14 = | .13+.001i .04+.65i  |
      | .8+.86i   .82+.5i   |
      | .97+.85i  .36+.028i |
      | .18+.16i  .097+.18i |
      | .8+.73i   .81+.57i  |
      | .56+.39i  .97+.08i  |
      | .059+.22i .36+.41i  |
      | .42+.42i  .93+.3i   |
      | .01+.78i  .68+.29i  |
      | .11+.7i   .32+.94i  |

                 10          2
o14 : Matrix CC     <--- CC
               53          53
i15 : x = solve(A,b)

o15 = | -.17-.6i  -.058-.34i |
      | .98+.24i  .7+1.5i    |
      | .34-.76i  .32-.39i   |
      | -.34+1.8i -.69+.33i  |
      | -.61-.35i -.12-1.1i  |
      | .45+.009i -.51-.13i  |
      | .26-.36i  1.2+.05i   |
      | -.24+.58i -.93-.42i  |
      | -.69-.62i .015-.26i  |
      | 1+.53i    .67+.98i   |

                 10          2
o15 : Matrix CC     <--- CC
               53          53
i16 : norm ( matrix A * matrix x - matrix b )

o16 = 9.99200722162641e-16

o16 : RR (of precision 53)
This may be used to invert a matrix over ZZ/p, RR or QQ.
i17 : A = random(RR^5, RR^5)

o17 = | .22 .81 .0058 .86  .65 |
      | .81 .55 .19   .37  .33 |
      | .31 .49 .24   .68  .12 |
      | .45 .89 .33   .057 .6  |
      | .32 .71 .64   .12  .39 |

                 5          5
o17 : Matrix RR    <--- RR
               53         53
i18 : I = id_(target A)

o18 = | 1 0 0 0 0 |
      | 0 1 0 0 0 |
      | 0 0 1 0 0 |
      | 0 0 0 1 0 |
      | 0 0 0 0 1 |

                 5          5
o18 : Matrix RR    <--- RR
               53         53
i19 : A' = solve(A,I)

o19 = | -.37 1.8  -.46 -.43 -.13 |
      | -2.1 -2.4 4.2  5    -3.4 |
      | .5   .55  -1.3 -2.7 3.3  |
      | .98  .41  .015 -1.8 .81  |
      | 3    1.9  -5.1 -3.7 3.3  |

                 5          5
o19 : Matrix RR    <--- RR
               53         53
i20 : norm(A*A' - I)

o20 = 3.33066907387547e-16

o20 : RR (of precision 53)
i21 : norm(A'*A - I)

o21 = 8.88178419700125e-16

o21 : RR (of precision 53)
Another method, which isn't generally as fast, and isn't as stable over RR or CC, is to lift the matrix b along the matrix A (see Matrix // Matrix).
i22 : A'' = I // A

o22 = | -.37 1.8  -.46 -.43 -.13 |
      | -2.1 -2.4 4.2  5    -3.4 |
      | .5   .55  -1.3 -2.7 3.3  |
      | .98  .41  .015 -1.8 .81  |
      | 3    1.9  -5.1 -3.7 3.3  |

                 5          5
o22 : Matrix RR    <--- RR
               53         53
i23 : norm(A' - A'')

o23 = 0

o23 : RR (of precision 53)

Caveat

This function is limited in scope, but is sometimes useful for very large matrices

See also

Ways to use solve :